Question
draw the following diagram: quadrilateral ABCD inscribed in circle E. the ⌒mAB = 54, ⌒mBC = 115, and ⌒mCD = 86
select the true statement
⌒mADC = 84.5
m/_D = 169
m/_B = 95.5
⌒mAD = 115
select the true statement
⌒mADC = 84.5
m/_D = 169
m/_B = 95.5
⌒mAD = 115
Answers
GPT-4o mini
To solve the problem, we first need to apply the properties of a circle and inscribed angles. We know the following:
1. The measures of the arcs:
- \( \text{arc } AB = 54^\circ \)
- \( \text{arc } BC = 115^\circ \)
- \( \text{arc } CD = 86^\circ \)
Using the property that the measure of an inscribed angle is half the measure of the intercepted arc, we can determine the angles.
### Step 1: Find the measure of arc \( AD \)
Since the total measure of a circle is \( 360^\circ \), we can find the measure of arc \( AD \):
\[
\text{arc } AD = 360^\circ - (\text{arc } AB + \text{arc } BC + \text{arc } CD)
\]
\[
\text{arc } AD = 360^\circ - (54^\circ + 115^\circ + 86^\circ) = 360^\circ - 255^\circ = 105^\circ
\]
### Step 2: Find angles \( D \) and \( B \)
- Angle \( D \) intercepts arc \( AB \) and \( AD \):
\[
m \angle D = \frac{1}{2}(\text{arc } AB + \text{arc } AD) = \frac{1}{2}(54^\circ + 105^\circ) = \frac{1}{2}(159^\circ) = 79.5^\circ
\]
- Angle \( B \) intercepts arc \( CD \) and \( BC \):
\[
m \angle B = \frac{1}{2}(\text{arc } BC + \text{arc } CD) = \frac{1}{2}(115^\circ + 86^\circ) = \frac{1}{2}(201^\circ) = 100.5^\circ
\]
### Conclusion: Analyze the Statements
1. \( \text{arc } AD = 105^\circ \) (not in options)
2. \( m \angle D = 79.5^\circ \) (not in options)
3. \( m \angle B = 100.5^\circ \) (not in options)
4. None of the provided statements match our calculations.
### Verification of Provided Statements:
- \( \text{arc } ADC = 84.5 \) (False, we found \( \text{arc } AD = 105^\circ \))
- \( m \angle D = 169 \) (False, we found \( m \angle D = 79.5^\circ \))
- \( m \angle B = 95.5 \) (False, we found \( m \angle B = 100.5^\circ \))
- \( \text{arc } AD = 115 \) (False, we found \( \text{arc } AD = 105^\circ \))
None of the statements provided are true based on the calculations performed above.
### Conclusion:
If you need an accurate diagram, it can be drawn as follows:
1. Draw a circle for circle \( E \).
2. Label points \( A, B, C, D \) on the circumference.
3. Indicate the measures of arcs \( AB \), \( BC \), and \( CD \) accordingly.
4. Based on the calculations, clarify that none of the statements provided are correct.
Let me know if you have other questions or require further assistance!
1. The measures of the arcs:
- \( \text{arc } AB = 54^\circ \)
- \( \text{arc } BC = 115^\circ \)
- \( \text{arc } CD = 86^\circ \)
Using the property that the measure of an inscribed angle is half the measure of the intercepted arc, we can determine the angles.
### Step 1: Find the measure of arc \( AD \)
Since the total measure of a circle is \( 360^\circ \), we can find the measure of arc \( AD \):
\[
\text{arc } AD = 360^\circ - (\text{arc } AB + \text{arc } BC + \text{arc } CD)
\]
\[
\text{arc } AD = 360^\circ - (54^\circ + 115^\circ + 86^\circ) = 360^\circ - 255^\circ = 105^\circ
\]
### Step 2: Find angles \( D \) and \( B \)
- Angle \( D \) intercepts arc \( AB \) and \( AD \):
\[
m \angle D = \frac{1}{2}(\text{arc } AB + \text{arc } AD) = \frac{1}{2}(54^\circ + 105^\circ) = \frac{1}{2}(159^\circ) = 79.5^\circ
\]
- Angle \( B \) intercepts arc \( CD \) and \( BC \):
\[
m \angle B = \frac{1}{2}(\text{arc } BC + \text{arc } CD) = \frac{1}{2}(115^\circ + 86^\circ) = \frac{1}{2}(201^\circ) = 100.5^\circ
\]
### Conclusion: Analyze the Statements
1. \( \text{arc } AD = 105^\circ \) (not in options)
2. \( m \angle D = 79.5^\circ \) (not in options)
3. \( m \angle B = 100.5^\circ \) (not in options)
4. None of the provided statements match our calculations.
### Verification of Provided Statements:
- \( \text{arc } ADC = 84.5 \) (False, we found \( \text{arc } AD = 105^\circ \))
- \( m \angle D = 169 \) (False, we found \( m \angle D = 79.5^\circ \))
- \( m \angle B = 95.5 \) (False, we found \( m \angle B = 100.5^\circ \))
- \( \text{arc } AD = 115 \) (False, we found \( \text{arc } AD = 105^\circ \))
None of the statements provided are true based on the calculations performed above.
### Conclusion:
If you need an accurate diagram, it can be drawn as follows:
1. Draw a circle for circle \( E \).
2. Label points \( A, B, C, D \) on the circumference.
3. Indicate the measures of arcs \( AB \), \( BC \), and \( CD \) accordingly.
4. Based on the calculations, clarify that none of the statements provided are correct.
Let me know if you have other questions or require further assistance!